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Stochastic methods for physics and biology, spring 2010

Lecturer

Stefan Geritz
Paolo Muratore-Ginanneschi

Scope

The aim of the course is to introduce the basic concepts of the theory of stochastic differential equations needed in applications (applied mathematics, physics and biology). In particular we will illustrate methods of qualitative, asymptotic and numerical analysis.

10 cu.

Type

Advanced studies.

Prerequisites

Lectures

Weeks 3-9 and 11-18, Monday 14-16 in room C124 and Friday 14-16 in room C123.

Easter holiday 1.-7.4.

First lecture: Friday, 22.01.2010

Lecture Notes

The lecture notes cover and sometimes integrate the material expounded in the lections. They also give bibliographic references for the same topics. For exercises please refer to the section below.

Lectures 1-10

Lectures 10-20

Lecture_01: Background of Probability

Lecture_11: Kolmogorov-Chentsov theorem

Lecture_02 (rev 29.01): Bernoulli variables, Borel-Cantelli lemma

Lecture_12: White-noise

Lecture_03: Limit theorems

Lecture_13: Ito integral

Lecture_04: Martingales

Lecture_14: Ito and Stratonovich calculus

Lecture_05: Numerical generation of random variables

Lecture_15: SDE, existence and uniqueness

Lecture_06 (rev 12.02): Statistical tests

Lecture_16: Kolmogorov pair and diffusion

Lecture_07: Fokker-Planck equation I

Lecture_17: Exit time statistics

Lecture_08: Fokker-Planck equation II

Lecture_18: Girsanov formula

Lecture_09: Brownian motion

Lecture_19: Numerical integration of SDE's

Lecture_10: Karhunen-Loève representation

Lecture_20: Population dynamics

Probabilistic interpretation of set theoretic concepts

Exams

There will be two options for taking the course. Passing an exam to be held after the end of the course, in
date to be agreed or working out a short project (recommended).

Bibliography

  1. L.C. Evans, "An Introduction to Stochastic Differential Equations", lecture notes.
  2. P. E. Kloeden, E. Platen, H. Schurz, "Numerical Solution Of Sde Through Computer Experiments", Springer (Universitext) (2003).
  3. N. Berglund, B. Gentz "Noise-Induced Phenomena in Slow-Fast Dynamical Systems. A Sample-Paths Approach", Springer (Probability and its Applications) (2005).
  4. D.J. Higham, "An algorithmic introduction to numerical simulation of stochastic differential equations" SIAM Review, Education Section, 43, 2001, 525-546. (Link to Higham's publications page.)

Book extended preview is available on line from google books

Registration

Did you forget to register? What to do.

Exercise groups

Group

Day

Time

Place

Instructor

1.

Wednesday

16-18

C124

 

Exercise set 1
Exercise set 2
Exercise set 3
Exercise set 4
Exercise set 5

Courseware

The codes below make use of the pgplot graphic library and of the GNU GSL
scientific library. You need to install these packages in order to use the codes as they are.

A c code to generate histograms and its initialization code

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